A190952 Largest integer k for which exp(k) < k^n, n>=3.

4, 8, 12, 16, 21, 26, 30, 35, 40, 45, 51, 56, 61, 67, 72, 78, 84, 89, 95, 101, 107, 113, 119, 125, 131, 137, 144, 150, 156, 163, 169, 175, 182, 188, 195, 201, 208, 214, 221, 228, 234, 241, 248, 254, 261, 268, 275, 282, 288, 295, 302, 309, 316, 323, 330, 337, 344, 351

Offset: 3

Author

Shel Kaphan, May 24 2011

Keywords

• nonn

Comments

n=3 is the starting index because exp(x)>x^n for all x>=0 when n=1,2.

Conjecture: There are floor((n+1)/log(n+1))-2 terms less than or equal to n. - Benedict W. J. Irwin, Jun 15 2016

Crossrefs

Cf. A088346 (Smallest integer k where exp(x)>x^n for all x>=k)

Cf. A190951 (Closest integer to the largest real x such that exp(x) = x^n)

Programs

Mathematica

a[n_] := Floor[E^-ProductLog[-1, -1/n]]; Table[a[n], {n, 3, 60}]

Formula

Conjecture: G.f.: Sum_{ j>=1 } (Sum_{ k>=1 } x^(j+floor((k+1)/log(k+1)))) + x^j. - Benedict W. J. Irwin, Jun 15 2016

a(n) = floor(-n*LambertW(-1,-1/n)). - Vaclav Kotesovec, Jun 29 2016